Fractions
A fraction represents a part of a whole. It is written as a⁄b, where a is the numerator (the number of parts we have) and b is the denominator (the total number of equal parts the whole is divided into).
LCM and HCF
Before working with fractions, we need to understand two important concepts: Lowest Common Multiple (LCM) and Highest Common Factor (HCF).
What is LCM?
The Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers.
Example: Finding LCM of 4 and 6
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 6: 6, 12, 18, 24, 30, ...
The smallest common multiple is 12
Answer: LCM(4,6) = 12
What is HCF?
The Highest Common Factor (HCF) is the largest number that divides exactly into two or more numbers.
Example: Finding HCF of 12 and 18
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
The largest common factor is 6
Answer: HCF(12,18) = 6
Uses: LCM helps us find common denominators when adding or subtracting fractions. HCF helps us simplify fractions to their lowest terms.
Equivalent Fractions
Equivalent fractions are fractions that represent the same value, even though they look different. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same number.
Example 1: Finding Equivalent Fractions
Find three fractions equivalent to ½.
Multiply numerator and denominator by 2: 2⁄4
Multiply numerator and denominator by 3: 3⁄6
Multiply numerator and denominator by 4: 4⁄8
Answer: ²⁄₄, ³⁄₆, ⁴⁄₈
Example 2: Simplifying Fractions (using HCF)
Simplify 12⁄18.
HCF of 12 and 18 is 6
Divide numerator and denominator by 6: 12÷6⁄18÷6 = 2⁄3
Answer: ¹²⁄₁₈ = ⅔
Identifying Fractions and Decimals
A fraction represents a part of a whole. A decimal is another way to represent a part of a whole using place value.
Example: Identifying Fractions from Shapes
If a circle is divided into 8 equal parts and 3 parts are shaded, what fraction is shaded?
Answer: 3⁄8 of the circle is shaded.
Example: Identifying Decimals
The digit 7 in 0.78 is in the tenths place (7 tenths = 0.7)
The digit 8 is in the hundredths place (8 hundredths = 0.08)
Answer: 0.78 = 7 tenths + 8 hundredths
Improper Fractions and Mixed Numbers
An improper fraction has a numerator larger than or equal to its denominator (e.g., ⁷⁄₄). A mixed number has a whole number part and a fractional part (e.g., 1¾).
Example 1: Converting Improper Fraction to Mixed Number
Convert ⁷⁄₄ to a mixed number.
7 ÷ 4 = 1 remainder 3
So, ⁷⁄₄ = 1¾
Answer: 1¾
Example 2: Converting Mixed Number to Improper Fraction
Convert 2⅗ to an improper fraction.
Whole number × denominator + numerator = (2 × 5) + 3 = 10 + 3 = 13
Answer: ¹³⁄₅
Adding and Subtracting Fractions
To add or subtract fractions, they must have the same denominator. If they don't, find the LCM of the denominators to create a common denominator.
Example 1: Adding Fractions with Same Denominator
2⁄5 + 1⁄5 = (2 + 1)⁄5 = 3⁄5
Answer: ³⁄₅
Example 2: Adding Fractions with Different Denominators
1⁄3 + 1⁄4
Step 1: Find the Least Common Multiple (LCM) of the denominators (3 and 4). The LCM is 12.
Step 2: Convert the fractions to have the denominator 12. To do this, multiply the top and bottom by the same number:
1 (×4)⁄3 (×4) = 4⁄12
1 (×3)⁄4 (×3) = 3⁄12
Step 3: Now that the denominators match, add the numerators together:
4⁄12 + 3⁄12 = (4 + 3)⁄12 = 7⁄12
Answer: 7⁄12
Example 3: Subtracting Fractions
5⁄6 - 1⁄3
Step 1: Find the Least Common Multiple (LCM) of 6 and 3. The LCM is 6.
Step 2: Convert the fractions to have the denominator 6. Since the first fraction already has 6, we only convert the second one:
1 (×2)⁄3 (×2) = 2⁄6
Step 3: Subtract the numerators while keeping the denominator the same:
5⁄6 - 2⁄6 = (5 - 2)⁄6 = 3⁄6
Step 4: Simplify the fraction by dividing both the top and bottom by their highest common factor (3):
3 (÷3)⁄6 (÷3) = 1⁄2
Answer: 1⁄2
Example 4: Adding Improper Fractions
7⁄4 + 5⁄4
Step 1: Since the denominators are already the same (4), you can add the numerators directly:
7⁄4 + 5⁄4 = (7 + 5)⁄4 = 12⁄4
Step 2: Simplify the fraction by dividing the numerator by the denominator:
12 ÷ 4 = 3
Answer: 3
Ordering Fractions
To arrange fractions in order of size, convert them to equivalent fractions with the same denominator (the LCM), then compare the numerators.
Example: Arranging Fractions in Ascending Order
Arrange 2⁄3, 3⁄5, 3⁄4 in ascending order (smallest to largest).
Step 1: Find the LCM of the denominators (3, 5, and 4).
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
The smallest common multiple is 60. So, LCM = 60.
Step 2: Convert each fraction to an equivalent fraction with denominator 60.
2⁄3 = (2 × 20)⁄(3 × 20) = 40⁄60
3⁄5 = (3 × 12)⁄(5 × 12) = 36⁄60
3⁄4 = (3 × 15)⁄(4 × 15) = 45⁄60
Step 3: Compare the numerators.
36⁄60 (smallest), 40⁄60, 45⁄60 (largest)
Step 4: Write the original fractions in ascending order.
3⁄5 , 2⁄3 , 3⁄4
Answer: 3⁄5, 2⁄3, 3⁄4
Dividing Fractions
To divide fractions, multiply by the reciprocal of the second fraction (flip the numerator and denominator of the divisor).
What is a Reciprocal?
The reciprocal of a fraction is obtained by swapping its numerator and denominator.
Example: Reciprocal of 2⁄3 is 3⁄2. Reciprocal of 5 is 1⁄5.
Example 1: Dividing Two Fractions
Calculate 3⁄4 ÷ 2⁄3
Step 1: Find the reciprocal of the second fraction (the divisor).
The second fraction is 2⁄3.
Its reciprocal is 3⁄2.
Step 2: Change the division sign to multiplication and multiply by the reciprocal.
3⁄4 ÷ 2⁄3 = 3⁄4 × 3⁄2
Step 3: Multiply the numerators together.
3 × 3 = 9
Step 4: Multiply the denominators together.
4 × 2 = 8
Step 5: Write the product as a fraction.
3⁄4 × 3⁄2 = (3 × 3)⁄(4 × 2) = 9⁄8
Step 6: Convert the improper fraction to a mixed number.
9 ÷ 8 = 1 remainder 1
9⁄8 = 11⁄8
Answer: 11⁄8
Example 2: Dividing a Fraction by a Whole Number
Calculate 5⁄6 ÷ 3
Step 1: Write the whole number as a fraction with denominator 1.
3 = 3⁄1
Step 2: Find the reciprocal of the second fraction (the divisor).
The second fraction is 3⁄1.
Its reciprocal is 1⁄3.
Step 3: Change the division sign to multiplication and multiply by the reciprocal.
5⁄6 ÷ 3⁄1 = 5⁄6 × 1⁄3
Step 4: Multiply the numerators together.
5 × 1 = 5
Step 5: Multiply the denominators together.
6 × 3 = 18
Step 6: Write the product as a fraction.
5⁄6 × 1⁄3 = (5 × 1)⁄(6 × 3) = 5⁄18
Answer: 5⁄18
Example 3: Dividing Mixed Numbers
Calculate 21⁄4 ÷ 11⁄2
Step 1: Convert each mixed number to an improper fraction.
21⁄4 = (2 × 4 + 1)⁄4 = 9⁄4
11⁄2 = (1 × 2 + 1)⁄2 = 3⁄2
Step 2: Find the reciprocal of the second fraction (the divisor).
The second fraction is 3⁄2.
Its reciprocal is 2⁄3.
Step 3: Change the division sign to multiplication and multiply by the reciprocal.
9⁄4 ÷ 3⁄2 = 9⁄4 × 2⁄3
Step 4: Multiply the numerators together.
9 × 2 = 18
Step 5: Multiply the denominators together.
4 × 3 = 12
Step 6: Write the product as a fraction.
9⁄4 × 2⁄3 = (9 × 2)⁄(4 × 3) = 18⁄12
Step 7: Simplify the fraction.
HCF of 18 and 12 is 6
18 ÷ 6⁄12 ÷ 6 = 3⁄2
Step 8: Convert the improper fraction to a mixed number.
3 ÷ 2 = 1 remainder 1
3⁄2 = 11⁄2
Answer: 11⁄2
Fractions and Decimals with Whole Numbers
When multiplying or dividing fractions and decimals by whole numbers greater than 10, the same rules apply. The numbers are larger, but the method does not change.
Example 1: Multiplying a Fraction by a Large Whole Number
Calculate 3⁄4 × 24
Step 1: Write the whole number as a fraction with denominator 1.
24 = 24⁄1
Step 2: Multiply the numerators together.
3 × 24 = 72
Step 3: Multiply the denominators together.
4 × 1 = 4
Step 4: Write the product as a fraction.
3⁄4 × 24 = (3 × 24)⁄(4 × 1) = 72⁄4
Step 5: Simplify the fraction by dividing the numerator by the denominator.
72 ÷ 4 = 18
Answer: 18
Example 2: Multiplying a Decimal by a Large Whole Number
Calculate 0.25 × 36
Step 1: Convert the decimal to a fraction.
0.25 = 25⁄100 = 1⁄4
Step 2: Multiply the fraction by the whole number.
1⁄4 × 36 = (1 × 36)⁄(4 × 1) = 36⁄4
Step 3: Simplify the fraction by dividing the numerator by the denominator.
36 ÷ 4 = 9
Answer: 9
Example 3: Dividing a Fraction by a Large Whole Number
Calculate 5⁄6 ÷ 15
Step 1: Write the whole number as a fraction with denominator 1.
15 = 15⁄1
Step 2: Find the reciprocal of the second fraction (the divisor).
The second fraction is 15⁄1.
Its reciprocal is 1⁄15.
Step 3: Change the division sign to multiplication and multiply by the reciprocal.
5⁄6 ÷ 15⁄1 = 5⁄6 × 1⁄15
Step 4: Multiply the numerators together.
5 × 1 = 5
Step 5: Multiply the denominators together.
6 × 15 = 90
Step 6: Write the product as a fraction.
5⁄6 × 1⁄15 = (5 × 1)⁄(6 × 15) = 5⁄90
Step 7: Simplify the fraction.
HCF of 5 and 90 is 5
5 ÷ 5⁄90 ÷ 5 = 1⁄18
Answer: 1⁄18
Example 4: Dividing a Decimal by a Large Whole Number
Calculate 4.8 ÷ 12
Step 1: Convert the decimal to a fraction.
4.8 = 48⁄10 = 24⁄5
Step 2: Write the whole number as a fraction with denominator 1.
12 = 12⁄1
Step 3: Find the reciprocal of the second fraction (the divisor).
The second fraction is 12⁄1.
Its reciprocal is 1⁄12.
Step 4: Change the division sign to multiplication and multiply by the reciprocal.
24⁄5 ÷ 12⁄1 = 24⁄5 × 1⁄12
Step 5: Multiply the numerators together.
24 × 1 = 24
Step 6: Multiply the denominators together.
5 × 12 = 60
Step 7: Write the product as a fraction.
24⁄5 × 1⁄12 = (24 × 1)⁄(5 × 12) = 24⁄60
Step 8: Simplify the fraction.
HCF of 24 and 60 is 12
24 ÷ 12⁄60 ÷ 12 = 2⁄5
Step 9: Convert the fraction to a decimal (if needed).
2⁄5 = 0.4
Answer: 0.4
Key Rules Summary:
• Multiply fractions: numerator × numerator, denominator × denominator
• Divide fractions: multiply by the reciprocal
• Add/subtract fractions: same denominator needed (use LCM)
• Simplify fractions: divide by HCF
• For decimals: convert to fractions first, or use place value understanding
NB: Always simplify your final answer. To find a common denominator, use the LCM of the denominators. To simplify a fraction, divide by the HCF of the numerator and denominator.